Hausdorff content at a prescribed scale

Let $(X,d)$ be a metric space, $A\subseteq X$, $s\in [0,\mathrm{\infty })$ a finite real, and $0<\delta \le \mathrm{\infty }$. A cover is a finite or countably infinite family $(U_j)$ of nonempty arbitrary subsets of $X$, with $A\subseteq {\bigcup }_j{U}_j$ and $\mathrm{diam}{U}_j\le \delta$. The empty family is permitted, and covers precisely the empty set. Define

${\mathcal{H}}_{\delta }^s(A)=\inf \{{\sum }_j(\mathrm{diam}{U}_j{)}^s:(U_j)\text{ is such a cover}\}\mathrm{.}$

Use Extended diameter for Hausdorff covers and the nonnegative extended sums of Series in the nonnegative extended real line. For this covering cost only, define $r^0=1$ for every $r\in [0,\mathrm{\infty }]$, including $0$ and $\mathrm{\infty }$; for $s>0$, use Real powers for positive bases, with the zero-base positive-exponent convention at finite bases and set ${\mathrm{\infty }}^s=\mathrm{\infty }$. Thus a nonempty singleton costs one when $s=0$, and zero when $s>0$. Finite covers are not padded with empty sets.

The infimum is in $[0,\mathrm{\infty }]$, with $\inf \mathrm{\varnothing }=\mathrm{\infty }$ when there is no admissible cover; existence follows from Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$. The empty sum is zero. The value ${\mathcal{H}}_{\mathrm{\infty }}^s$ is called Hausdorff content; ${\mathcal{H}}_{\delta }^s$ for finite $\delta$ is the scale approximation.